1. Let F(x, y, z) = (x cos y)i + (yx)j + (z sin y)k be...

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1. Let F(x, y, z) = (x cos² y)i + (yx)j + (z sin² y)k be a vector field on R³. Let ƒ : R³ → R be a function defined by f(x, y, z) = et + z³ cos (xy). (a) Find the gradient of f. (b) Find the divergence and curl of F. Fully simplify your answers. 2. Let F(x, y, z) = (xy)i + (yz)j + (xz)k be a vector field on R³. Verify by computation, not using a theorem, that V (V x F) = 0, i.e. that the divergence of the curl of Fis zero. 1. Let F(x, y, z) = (x cos² y)i + (yx)j + (z sin² y)k be a vector field on R³. Let ƒ : R³ → R be a function defined by f(x, y, z) = et + z³ cos (xy). (a) Find the gradient of f. (b) Find the divergence and curl of F. Fully simplify your answers. 2. Let F(x, y, z) = (xy)i + (yz)j + (xz)k be a vector field on R³. Verify by computation, not using a theorem, that V (V x F) = 0, i.e. that the divergence of the curl of Fis zero.